Resonance and strong resonance for semilinear elliptic equations in ℝN
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We prove the existence of weak solutions for the semilinear elliptic problem
-Δu = λhu + αg(u) + ƒ, u ∈ D1,2 (ℝN),
where λ ∈ ℝ, ƒ ∈ L2N/ (N+2), g : ℝ → ℝ is a continuous bounded function, and h ∈ LN/2 ⋂ Lα, α > N/2. We assume that α ∈ L1 ⋂ L∞ and ƒ ≡ 0 for the case of strong resonance. We prove first that the Palais-Smale condition holds for the functional associated with the semilinear problem using the concentration-compactness lemma of Lions. Then we prove the existence of weak solutions by applying the saddle point theorem of Rabinowitz for the cases of non-resonance and resonance, and a linking theorem of Silva in the case of strong resonance. The main theorems in this paper constitute an extension to ℝN of previous results in bounded domains by Ahmad, Lazar, and Paul , for the case of resonance, and by Silva  in the strong resonance case.
CitationLopez Garza, G., & Rumbos, A. J. (2003). Resonance and strong resonance for semilinear elliptic equations in ℝN. Electronic Journal of Differential Equations, 2003(124), pp. 1-22.
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